Download Construction of P-box

Structural reliability analysis with probabilityboxes
Hao Zhang
School of Civil Engineering, University of Sydney, NSW 2006, Australia
Michael Beer
Institute for Risk & Uncertainty, University of Liverpool, Liverpool, UK
Reliability assessment with limited data
A common scenario





Available data on structural strength and loads are
typically limited.
Difficulty in identifying the distribution (type,
parameters).
Competing probabilistic models.
Tail sensitivity.
Choice of probabilistic model is epistemic in
nature.
2
Reliability assessment with limited data
Options for solution

Bayesian approach



more subjective
high numerical effort
Imprecise probabilities



Probability box
Random set
Dempster-Shafer evidence theory
3
Presentation outline

Quasi interval Monte Carlo method

Different approaches for constructing P-boxes

Example
4
Monte Carlo method

Probability of failure, Pf , is estimated by

Inverse transform method
rj : a sample of iid standard uniform random variates.
5
Interval Monte Carlo method
When Fx( ) is unknown but bounded, interval samples can be
generated
Define
then
One has
6
Interval Monte Carlo method
A lower and an upper bound for Pf can be estimated as
Variance of direct interval Monte Carlo
7
Low-discrepancy sequences
Improvement of
- sampling quality
- convergence
- numerical efficiency
2D scatter plots: (a) random sample; (b) Good lattice point; (c) Halton
sequence; (d) Faure sequence.
8
Variance for interval quasi-Monte Carlo
A variance-type error estimate cannot be obtained directly
because low-discrepancy sequences are deterministic.

An empirical variance estimate for interval quasi-Monte
Carlo can be obtained by using randomized low-discrepancy
sequence.

9
Presentation outline

Quasi interval Monte Carlo method

Different approaches for constructing P-boxes

Example
10
Construction of P-box
Kolmogorov-Smirnov confidence limits
Fn(x) = empirical cumulative frequency function
Dnα = K-S critical value at significance level α with a sample size of n


Non-parametric.
The derived p-box has to be truncated.
11
Construction of P-box
Chebyshev’s inequality
If the knowledge of the first two moments (and the range) of the random
variable is available, (one-sided or two-sided) Chebyshev inequality can
be used.


Non-parametric.
Independent of sample size.
12
Construction of P-box
Distributions with interval parameters
If the (unknown) statistical parameter (θ ) of the distribution varies in an
interval


Parametric representation.
Confidence intervals on statistics provide a natural way to define
interval bounds of the distribution parameters.
13
Construction of P-box
Envelope of competing probability models
When there are multiple candidate distribution models which cannot be
distinguished by standard goodness-of-fit tests,
Fi (x) = ith candidate CDF
14
Presentation outline

Quasi interval Monte Carlo method

Different approaches for constructing P-boxes

Example
15
Example
Limit state: roof drift < 17.78 mm
Roof drift is computed by (linear elastic) finite
element analysis.
10-bar truss (after Nie and Ellingwood, 2005)
16
Example



The K-S limit and Chebyshev bound are truncated at 50 kN and 220 kN.
Type 1 Largest distribution with interval mean ([100.28, 125.69] kN, 95%
confidence interval)
Five candidate distributions: T1 Largest, lognormal, Gamma, Normal, and
Weibull, which all pass the K-S tests at a significance level of 5%.
17
Example
18
Discussion
K-S approach





K-S p-box yields a very wide reliability bound ([0, 0.246]).
The K-S wind load p-box itself is very wide, particularly in
its upper tail.
K-S p-box has to be truncated at the tails.
The truncation points are often chosen arbitrarily.
The result may be influenced strongly by the truncation.
19
Discussion
Chebyshev inequality





One-sided Chebyshev p-box yields a very wide reliability
bound ([0, 0.103]).
It also has the truncation problem.
Chebyshev inequality is independent of the sample size.
Two sets of data, one with limited samples and a second
with comprehensive samples, would lead to the same pbox if they have the same first 2 moments.
General conception: epistemic uncertainty can be reduced
when the quality of data is refined.
20
Discussion
Distribution with interval parameters



Pf varies between 0.0116 and 0.0266.
This interval bound clearly demonstrates the effect of
small sample size on the calculated failure probability.
It appears that confidence intervals on distribution
parameters is a reasonable way to define p-box, provided
that the appropriate distribution form can be discerned.
21
Discussion
Envelope of candidate distributions




Pf varies between 0.0006 and 0.0162.
The lower bound of Pf is contributed by the Weibull
distribution.
If Weibull is discarded, the bounds of Pf will be
[0.0032, 0.0162].
These results highlight the sensitivity of the failure
probability to the choice of the probabilistic model for the
wind load.
22
Conclusions




Interval quasi-Monte Carlo method is efficient and
its implementation is relatively straightforward.
A truss structure has been analysed.
Reliability bounds based on different wind load pbox models vary considerably.
Failure probabilities are controlled by the tails of
the distributions.
23
Conclusions


Both K-S confidence limits and Chebyshev
inequality have shown some practical difficulties
to define p-boxes in the context of structural
reliability analysis (tail sensitivity problem).
The most reasonable method to construct p-box
for the purpose of reliability assessment seems to
be their construction based on confidence intervals
of statistics.
24